Semisimple weak Hopf algebras

We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S-2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S-2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the categorical and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zhu is established and properties of A-module algebras and their dimensions are studied. (C) 2004 Elsevier Inc. All rights reserved.